Am I dry if I pass the drop rate?

Not by much. At exactly the drop rate (say 5,000 kills on a 1/5,000), about 37% of players still have nothing. At twice the rate it is about 13.5%, and at three times about 5%.

Does killing more make the next drop more likely?

No. Each kill is a fresh roll at the same rate. Past kills only change how unlikely your streak was.

Drop rate and dry calculator

Enter a drop rate and your kill count. Each kill is an independent roll, so the chance of still having nothing after n kills is (1 − 1/rate)ⁿ.

Drop rate: 1 in

Enter 5000 for a 1/5,000 drop

Kills or attempts so far

Chance of at least one drop by now86.47%

Chance of still having none13.53%

How dry2.00× the drop rate

Kills for a 50% chance3,466

Kills for a 90% chance11,512

Only 13.53% of players are this dry or drier. Every kill is still a fresh 1 in 5,000.

Questions

Am I dry if I pass the drop rate?: Not by much. At exactly the drop rate (say 5,000 kills on a 1/5,000), about 37% of players still have nothing. At twice the rate it is about 13.5%, and at three times about 5%.
Does killing more make the next drop more likely?: No. Each kill is a fresh roll at the same rate. Past kills only change how unlikely your streak was.